This paper focuses on estimation of high-dimensional bandable covariance matrices and sparse covariance matrices in the presence of missing data. These two classes of covariance matrices arise frequently in many applications, including genomics, econometrics, signal processing, temporal and spatial data analyses, and chemometrics.
Estimation of these high-dimensional structured covariance matrices have been well studied in the setting of complete data in a number of recent papers, e.g., Bickel and Levina [1]}, [2]}, Karoui [3]}, Rothman et al. [4]}, Cai and Zhou [5]}, Cai and Liu [6]}, Cai et al. [7]}, [8]} and Cai and Yuan [9]}. Given an incomplete sample \({\mathbf {X}}^* \)  with missing values, we introduced a “generalized" sample covariance matrix, which can be viewed as an analog of the usual sample covariance matrix in the case of complete data. For estimation of bandable covariance matrices, where the entries of the matrix decay as they move away from the diagonal, a blockwise tridiagonal estimator is introduced and is shown to be rate-optimal. We then consider estimation of sparse covariance matrices. An adaptive thresholding estimator based on the generalized sample covariance matrix is proposed. The estimator is shown to achieve the optimal rate of convergence over a large class of approximately sparse covariance matrices under mild conditions.
